Subcritical and supercritical granular flow around an obstacle on a rough inclined plane

A blunt obstacle in the path of a rapid granular avalanche generates a bow shock (a jump in the avalanche thickness and velocity), a region of static grains upstream of the obstacle, and a grain-free region downstream. Here, it is shown that this interaction is qualitatively altered if the incline on which the avalanche is flowing is changed from smooth to rough. On a rough incline, the friction between the grains and the incline depends on the flow thickness and speed, which allows both rapid (supercritical) and slow (subcritical) steady uniform avalanches. For supercritical experimental flows, the material is diverted around a blunt obstacle by the formation of a bow shock and a static dead zone upstream of the obstacle. Downslope, a grain-free vacuum region forms, but, in contrast to flows on smooth beds, static levees form at the boundary between the vacuum region and the flow. In slower, subcritical, flows the flow is diverted smoothly around the dead zone and the obstacle without forming a bow shock. After the avalanche stops, signatures of the dead zone, levees and (in subcritical flows) a deeper region upslope of the obstacle are frozen into the deposit. To capture this behaviour, numerical simulations are performed with a depth-averaged avalanche model that includes frictional hysteresis and depth-averaged viscous terms, which are needed to accurately model the flowing and deposited regions. These results may be directly relevant to geophysical mass flows and snow avalanches, which flow over rough terrain and may impact barriers or other infrastructure.

Spontaneous formation of waves, channels and levees in geophysical mass flows

<p>Geophysical mass flows often break down into large amplitude wave pulses and/or spontaneously form channels with static levees in the arrest zone, enhancing overall run-out. This talk reviews recent depth-averaged models that are able to capture the formation of:- (i) rollwaves, (ii) erosion-deposition waves (which exchange mass with an erodible substrate) and (iii) channel and levee formation, within a single framework. The key is the inclusion of frictional hysteresis, which allows static and moving zones to coexist, as well as depth-averaged viscous terms that incorporate further details of the granular rheology. As well as being able to compute time-dependent spatially evolving solutions numerically, the resulting model allows steady-state solutions to be constructed for the height, width and depth-averaged velocity profile across a leveed channel, which are in good quantitative agreement with small scale analogue experiments using monodisperse dry sand. Colour change experiments are used to show that erosion-deposition waves really do propagate downslope as a wave, rather than a coherent body of grains, and that the presence of the erodible substrate gives them surprising mobility over very long distances. Photos and videos of the similar effects at field scale will be shown to emphasize the importance of these ideas for a wide range of geophysical mass flows. There are, however, still many open challenges in how to generalize these results to multiphase mixtures with broad grain size distributions.</p>

Frictional hysteresis and particle deposition in granular free-surface flows

Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steady-uniform granular flow is brought to rest it leaves a deposit of thickness $h_{stop}(\unicode[STIX]{x1D701})$ on a rough slope inclined at an angle $\unicode[STIX]{x1D701}$ to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle $\unicode[STIX]{x1D701}_{start}$, or the thickness is increased to $h_{start}(\unicode[STIX]{x1D701})>h_{stop}(\unicode[STIX]{x1D701})$. This simple phenomenology leads to a rich variety of flows with co-existing regions of solid-like and fluid-like granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of self-channelized flows with static levees, retrogressive failures as well as erosion–deposition waves that travel through the material. This paper is motivated by the experimental observation that a travelling-wave develops, when a steady uniform flow of carborundum particles on a bed of larger glass beads, runs out to leave a deposit that is approximately equal to $h_{stop}$. Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278–315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steady-uniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness $h_{\ast }$) with the deposited layer. The first wave moves slightly faster than the second wave, and so there is a slowly expanding region between them in which the flow thins and the particles slow down. An exact inviscid solution for the second travelling wave is derived and it is shown that for a steady-uniform flow of thickness $h_{\ast }$ it produces a deposit close to $h_{stop}$ for all inclination angles. Numerical simulations show that the two-wave structure deposits layers that are approximately equal to $h_{stop}$ for all initial thicknesses. This insensitivity to the initial conditions implies that $h_{stop}$ is a universal quantity, at least for carborundum particles on a bed of larger glass beads. Numerical simulations are therefore able to capture the complete experimental staircase procedure, which is commonly used to determine the $h_{stop}$ and $h_{start}$ curves by progressively increasing the inclination of the chute. In general, however, the deposit thickness may depend on the depth of the flowing layer that generated it, so the most robust way to determine $h_{stop}$ is to measure the deposit thickness from a flow that was moving at the minimum steady-uniform velocity. Finally, some of the pathologies in earlier non-monotonic friction laws are discussed and it is explicitly shown that with these models either steadily travelling deposition waves do not form or they do not leave the correct deposit depth $h_{stop}$.

Retrogressive failure of a static granular layer on an inclined plane

When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwardset al.(J. Fluid. Mech., vol. 823, 2017, pp. 278–315,J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.

Frictional Hysteresis Model for Stick–Slip Behavior of Magnetorheological Elastomer Under Various Magnetic Field Strengths

In recent studies, many mathematical models have been introduced to describe the shear deformation characteristics of a magnetorheological elastomer (MRE). Owing to its beneficial elastomeric characteristics, an MRE can be adopted in novel controllable devices such as friction dampers and brakes. In this study, mathematical models are introduced to identify the frictional behavior of an MRE under different magnetic field conditions. Specifically, the improved LuGre (I-LuGre) model and the strain-stiffening model are compared using a system identification method. To identify the model that best describes the stick/slip behavior of an MRE, a harmonic frictional force was exerted on its surface with magnetic fields of varying strength. The I-LuGre model showed a precise correlation with the experimental results, and the strain-stiffening model was shown to have a simple structure for describing the frictional phenomenon. The system output error of the I-LuGre model remained within smaller errors than that of the strain-stiffening model. The parameter variations of each model that can be utilized to construct a control strategy are provided herein.

Spiral Power Springs. Part 2—Design

The theoretical model developed in a preceding companion paper for predicting the torque-turn behavior of a spiral power spring is extended in an effort to improve the practical applicability. A theoretical equation based on springback theory is developed for computing the shape factor in the logarithmic spiral and an empirical equation for calculating the magnitude of frictional hysteresis is obtained. The extended model is tested with two springs and the experimental and calculated results are in good accord. The computation procedure involved in practical application of the extended model for spring design is discussed.